Minimum distance from corners

Geometry Level 3

Four points are located on corner of a square of length 1 km. Government wants to build a road connecting all 4 points.

Is the following statement True or False ?

Minimum length of road required to connect all 4 points is $2\sqrt 2 \approx 2.828$ km.

True False

1 solution

Manish Bisarya
Nov 22, 2018

As per solution to Steiner problem, ∠AED = 120°.

Let Point G be midway between A & D.

∴ AG = $\frac {1}{2}$ & ∠AEG = 60°

∴ AE = DE = BF = CF = AG $\times$ cosec 60° = $\frac {1}{2}$ $\times$ $\frac {2}{\sqrt{3}}$ = $\frac {1}{\sqrt{3}}$

∴ GE = AG $\times$ cot 60° = $\frac {1}{ 2 \sqrt{3}}$

⇒ EF = 1 - (2 $\times$ GE) = 1 - (2 $\times$ $\frac {1}{2 \sqrt{3}}$ ) = 1 - $\frac {1}{\sqrt{3}}$

∴ Total Length = 4 AE + EF = (4 $\times$ $\frac {1}{\sqrt{3}}$ ) + (1 - $\frac {1}{\sqrt{3}}$ ) = $\frac {3}{\sqrt{3}}$ + 1 ≈ 2.732 KMs